3.2840 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{31704544 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{66706983 \sqrt{33}}+\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{5 x+3}}{733776813 \sqrt{3 x+2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac{940 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{9/2}}-\frac{924247516 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}} \]

[Out]

(940*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(9/2)) - (251590*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2139291*(2 +
3*x)^(7/2)) - (362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14975037*(2 + 3*x)^(5/2)) + (11460644*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(104825259*(2 + 3*x)^(3/2)) + (924247516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(733776813*Sqrt[2 + 3*x]) + (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) - (924247516*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(66706983*Sqrt[33]) - (31704544*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33]
)

________________________________________________________________________________________

Rubi [A]  time = 0.0958818, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{5 x+3}}{733776813 \sqrt{3 x+2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac{940 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{9/2}}-\frac{31704544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}}-\frac{924247516 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(13/2)),x]

[Out]

(940*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(9/2)) - (251590*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2139291*(2 +
3*x)^(7/2)) - (362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14975037*(2 + 3*x)^(5/2)) + (11460644*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(104825259*(2 + 3*x)^(3/2)) + (924247516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(733776813*Sqrt[2 + 3*x]) + (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) - (924247516*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(66706983*Sqrt[33]) - (31704544*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33]
)

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{13/2}} \, dx &=\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{2}{231} \int \frac{\left (-540-\frac{1855 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{4 \int \frac{-\frac{325685}{4}-\frac{551425 x}{4}}{\sqrt{1-2 x} (2+3 x)^{9/2} \sqrt{3+5 x}} \, dx}{43659}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{8 \int \frac{-\frac{3890945}{8}-\frac{3144875 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{2139291}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{16 \int \frac{-\frac{23392455}{8}-\frac{13599975 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{74875185}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{32 \int \frac{-\frac{868793295}{16}+\frac{214887075 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1572378885}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{3+5 x}}{733776813 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{64 \int \frac{-\frac{11051690775}{16}-\frac{17329640925 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11006652195}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{3+5 x}}{733776813 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}+\frac{15852272 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{66706983}+\frac{924247516 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{733776813}\\ &=\frac{940 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{9/2}}-\frac{251590 \sqrt{1-2 x} \sqrt{3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac{362666 \sqrt{1-2 x} \sqrt{3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac{11460644 \sqrt{1-2 x} \sqrt{3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac{924247516 \sqrt{1-2 x} \sqrt{3+5 x}}{733776813 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac{924247516 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}}-\frac{31704544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{66706983 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.302438, size = 112, normalized size = 0.45 \[ \frac{-6417960640 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (112296073194 x^5+377569336554 x^4+507518001945 x^3+340525216341 x^2+113962415157 x+15211411193\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17610643512 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(13/2)),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(15211411193 + 113962415157*x + 340525216341*x^2 + 507518001945*x^3 + 3775693
36554*x^4 + 112296073194*x^5))/(2 + 3*x)^(11/2) + 14787960256*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 6417960640*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(17610643512*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.025, size = 599, normalized size = 2.4 \begin{align*} -{\frac{2}{22013304390\,{x}^{2}[phone]\,x-6603991317} \left ( 112296073194\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-48736388610\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-162454628700\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-216606171600\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-144404114400\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3368882195820\,{x}^{7}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-48134704800\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-11663968316202\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -6417960640\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -15347583409266\,{x}^{5}-8340186467079\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+136902700737 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)/(2+3*x)^(13/2)/(1-2*x)^(1/2),x)

[Out]

-2/2201330439*(112296073194*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^5*(3+5*x)^(1/2)*(2+3*x)^
(1/2)*(1-2*x)^(1/2)-48736388610*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^5*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(2+3*x)^(1/2)
*(1-2*x)^(1/2)*(3+5*x)^(1/2)-162454628700*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(2+3*x)^
(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-216606171600*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^
3*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2
))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-144404114400*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66
^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-3368882195820*x^7+110909701920*2^(1/2)*EllipticE(1/11*(6
6+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-48134704800*2^(1/2)*EllipticF(1/11*
(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-11663968316202*x^6+14787960256*2^
(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-6417960640*2^(
1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-15347583409266*
x^5-8340186467079*x^4+127213913772*x^3+2266497365808*x^2+980027502834*x+136902700737)*(1-2*x)^(1/2)*(3+5*x)^(1
/2)/(10*x^2+x-3)/(2+3*x)^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(13/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(13/2)*sqrt(-2*x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(13/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 2
4948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(5/2)/(2+3*x)**(13/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(13/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(13/2)*sqrt(-2*x + 1)), x)